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Fluid Mechanics

The Reynolds Number, Inertia over Viscosity, Splits Laminar from Turbulent

Reynolds number Re = ρvD/μ (inertia/viscosity); critical Re~2300 splits laminar from turbulent; dimensionless

The same water sometimes flows smooth and sometimes churns into a tangle. Drag the Re slider to watch the flow flip, and see what decides between the two.

Place a small post in the flow and drag Re. At low Re the water glides smoothly around the post and the wake behind is calm; this orderly, layered flow is laminar. Raise Re and vortices peel off behind the post one after another into a churning mess; this is turbulent.

Whether flow is laminar or turbulent is settled by a tug-of-war between two forces. Inertia pushes any disturbance onward and grows it; viscosity smooths it over and damps it out. Re is exactly inertia divided by viscosity. Drag Re and see which side wins. Inertia winning means turbulent; viscosity winning means laminar.

The change is abrupt. Raise Re slowly. The moment it crosses a critical value, the smooth flow begins to break up. For pipe flow that boundary is around Re 2300. Below it is laminar, above it turbulent, and in between is a transitional band where it flickers between the two.

As an equation, Re = ρvDμ. Density times speed times size, divided by viscosity, a dimensionless number. Drag v and Re moves with it, crossing the critical line. Faster, fatter, thinner fluid all raise Re and tip it toward turbulent. Being a pure number with no units, it compares a river, a blood vessel, and an oil pipeline on the same scale.

Picture a kitchen faucet. Open it just a little and a smooth, clear column of water falls like a glass rod; that is laminar. Drag the flow up and at some point the column starts to bubble and break into white churn; it has crossed into turbulent. The same goes for incense smoke, rising straight at first and splitting up higher. In the next lesson we follow how this viscosity shapes the velocity profile inside a pipe, through the laminar parabola.

In PracticeTo sum up: whether a flow is laminar or turbulent is settled by the tug-of-war between inertia and viscosity, and their ratio is the Reynolds number Re = ρvDμ. When Re is small, viscosity wins and the flow is smooth and laminar; when it is large, inertia wins and the flow is a turbulent tangle. The critical Re for pipe flow is about 2300. Faster, fatter, thinner fluid all raise Re. Being dimensionless, it compares flows of wildly different size and speed on a single scale, and makes scale-model testing possible, the seed of the final lesson on dimensional analysis. In the next lesson we follow the parabolic laminar velocity profile that viscosity shapes inside a pipe.
Fluid Mechanics
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